Subgroups — C₂×C₂₃×#3354 #19208873

Order 5888.

2146 conjugacy classes of subgroups.

The subgroup lattice has 560 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#555	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₂₃×#547	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#3354	C₂	1	16	yes	maximal, direct product factor
4	C₂×C₂₃×#498	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₂₃×#481	C₂	1	2	yes	maximal, complemented
6	C₂×C₂₃×#482	C₂	1	2	yes	maximal, complemented
7	C₂×C₂₃×#476	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₂₃×#474	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₂₃×#469	C₂	1	2	yes	maximal, complemented
10	C₂×C₄×D₈×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₂₃×#436	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₂₃×#428	C₂	1	2	yes	maximal, complemented
13	C₂²×#41×C₂₃	—	2	1	no	—
14	C₂²×D₈×C₂₃	C₂²	1	1	yes	characteristic
15	C₂²×M₁₆×C₂₃	C₂²	1	2	yes	—
16	C₂₃×#555	—	2	4	no	—
17	C₂₃×#555	C₂²	1	8	yes	complemented
18	C₂₃×#547	—	2	4	no	—
19	C₂₃×#547	C₂²	1	8	yes	complemented
20	C₂×C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
21	C₂×C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
22	C₂×C₂₃×#124	—	2	2	no	—
23	C₂×C₂₃×#123	—	2	2	no	—
24	C₂×C₂₃×C₂²⋊Q₈	—	2	1	no	—
25	C₂×C₂₃×#121	—	2	1	no	—
26	C₂×C₂₃×#121	C₂²	1	2	yes	—
27	C₂×C₂₃×#121	C₂²	1	2	yes	—
28	C₂×C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
29	C₂×C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
30	C₂×C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
31	C₂×C₂₃×#117	—	2	1	no	—
32	C₂×C₂₃×#117	C₂²	1	2	yes	—
33	C₂²×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
34	C₂₃×#498	C₂²	1	8	yes	complemented
35	C₂₃×#482	C₂²	1	16	yes	complemented
36	C₂₃×#481	C₂²	1	16	yes	complemented
37	C₂₃×#476	C₂²	1	8	yes	complemented
38	C₂₃×#474	C₂²	1	8	yes	complemented
39	C₂₃×#469	C₂²	1	16	yes	complemented
40	C₄×D₈×C₂₃	C₂²	1	8	yes	complemented
41	C₂₃×#436	C₂²	1	8	yes	complemented
42	C₂₃×#428	C₂²	1	16	yes	complemented
43	C₂×C₂₃×#107	C₂²	1	1	yes	characteristic
44	C₂×C₂₃×#107	C₂²	1	2	yes	—
45	C₂×C₂₃×#106	C₂²	1	2	yes	—
46	C₂×C₂₃×#106	C₂²	1	2	yes	—
47	C₂×C₂₃×#106	C₂²	1	2	yes	—
48	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
49	C₂×C₂₃×#102	C₂²	1	1	yes	characteristic
50	C₂×C₂₃×#102	C₂²	1	1	yes	characteristic
51	C₂×C₂₃×#102	C₂²	1	2	yes	—
52	C₂×C₂₃×#102	C₂²	1	2	yes	—
53	C₂×C₂₃×C₂²⋊C₈	C₂²	1	2	yes	—
54	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
55	C₂×#41×C₂₃	—	2	8	no	—
56	C₂×#41×C₂₃	—	4	8	no	—
57	C₂²×Q₈×C₂₃	—	2	1	no	—
58	C₂²×Q₈×C₂₃	—	2	1	no	—
59	C₂²×D₄×C₂₃	—	2	2	no	—
60	C₂²×D₄×C₂₃	—	2	2	no	—
61	C₂²×D₄×C₂₃	D₄	1	1	yes	characteristic
62	C₂²×D₄×C₂₃	C₂³	1	1	yes	characteristic
63	C₂²×D₄×C₂₃	C₂³	1	1	yes	characteristic
64	C₂³×C₄×C₂₃	—	2	2	no	—
65	C₂³×C₄×C₂₃	—	2	1	no	—
66	C₂³×C₄×C₂₃	—	2	1	no	—
67	C₂³×C₄×C₂₃	C₂³	1	2	yes	—
68	C₂×D₈×C₂₃	—	2	4	no	—
69	C₂×D₈×C₂₃	C₂³	1	4	yes	—
70	C₂×M₁₆×C₂₃	—	2	8	no	—
71	C₂×M₁₆×C₂₃	C₂³	1	8	yes	—
72	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
73	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
74	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
75	C₂₃×C₄⋊Q₈	—	2	2	no	—
76	C₂₃×C₄⋊Q₈	—	2	2	no	—
77	C₂₃×C₄⋊Q₈	C₂³	1	4	yes	—
78	C₂₃×C₄⋊Q₈	C₂³	1	4	yes	—
79	C₂₃×#124	—	2	16	no	—
80	C₂₃×#123	—	2	16	no	—
81	C₂₃×C₂²⋊Q₈	—	2	8	no	—
82	C₂₃×#121	—	2	8	no	—
83	C₂₃×#121	—	2	4	no	—
84	C₂₃×#121	—	2	4	no	—
85	C₂₃×#121	C₂³	1	8	yes	—
86	C₂₃×#121	C₂³	1	8	yes	—
87	C₄×D₄×C₂₃	—	2	2	no	—
88	C₄×D₄×C₂₃	—	2	2	no	—
89	C₄×D₄×C₂₃	D₄	1	4	yes	—
90	C₄×D₄×C₂₃	C₂³	1	4	yes	—
91	C₄×D₄×C₂₃	C₂³	1	4	yes	—
92	C₄×D₄×C₂₃	C₂³	1	4	yes	—
93	C₂₃×#117	—	2	4	no	—
94	C₂₃×#117	—	2	8	no	—
95	C₂₃×#117	C₂³	1	8	yes	—
96	C₂×C₄⋊C₄×C₂₃	—	2	4	no	—
97	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
98	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
99	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
100	C₂×C₄⋊C₄×C₂₃	D₄	1	1	yes	characteristic
101	C₂×C₄⋊C₄×C₂₃	C₂³	1	8	yes	—
102	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
103	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
104	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
105	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
106	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
107	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
108	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
109	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
110	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
111	C₂×C₂²⋊C₄×C₂₃	D₄	1	2	yes	—
112	C₂×C₄²×C₂₃	—	2	2	no	—
113	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
114	C₂₃×#107	C₂³	1	4	yes	—
115	C₂₃×#107	C₂³	1	8	yes	—
116	C₂₃×#106	C₂³	1	8	yes	—
117	C₂₃×#106	C₂³	1	8	yes	—
118	C₂₃×#106	C₂³	1	8	yes	—
119	C₂₃×C₄⋊C₈	C₂³	1	4	yes	—
120	C₂₃×#102	C₂³	1	4	yes	—
121	C₂₃×#102	C₂³	1	4	yes	—
122	C₂₃×#102	C₂³	1	8	yes	—
123	C₂₃×#102	C₂³	1	8	yes	—
124	C₂₃×C₂²⋊C₈	C₂³	1	8	yes	—
125	C₄×C₈×C₂₃	C₂³	1	4	yes	—
126	C₂⁴×C₂₃	—	2	1	no	—
127	C₂⁴×C₂₃	—	2	1	no	—
128	C₂⁴×C₂₃	C₂×D₄	1	2	yes	—
129	#41×C₂₃	—	4	32	no	—
130	C₂×Q₈×C₂₃	—	2	4	no	—
131	C₂×Q₈×C₂₃	—	2	4	no	—
132	C₂×Q₈×C₂₃	—	4	4	no	—
133	C₂×Q₈×C₂₃	—	8	2	no	—
134	C₂×D₄×C₂₃	—	2	8	no	—
135	C₂×D₄×C₂₃	—	2	8	no	—
136	C₂×D₄×C₂₃	—	2	4	no	—
137	C₂×D₄×C₂₃	—	2	1	no	—
138	C₂×D₄×C₂₃	—	2	1	no	—
139	C₂×D₄×C₂₃	—	2	2	no	—
140	C₂×D₄×C₂₃	—	2	2	no	—
141	C₂×D₄×C₂₃	—	2	2	no	—
142	C₂×D₄×C₂₃	—	2	2	no	—
143	C₂×D₄×C₂₃	—	4	8	no	—
144	C₂×D₄×C₂₃	—	8	4	no	—
145	C₂×D₄×C₂₃	C₂×D₄	1	4	yes	—
146	C₂×D₄×C₂₃	C₂⁴	1	2	yes	—
147	C₂×D₄×C₂₃	C₂⁴	1	2	yes	—
148	C₂²×C₄×C₂₃	—	2	4	no	—
149	C₂²×C₄×C₂₃	—	2	4	no	—
150	C₂²×C₄×C₂₃	—	2	8	no	—
151	C₂²×C₄×C₂₃	—	2	1	no	—
152	C₂²×C₄×C₂₃	—	2	2	no	—
153	C₂²×C₄×C₂₃	—	2	1	no	—
154	C₂²×C₄×C₂₃	—	2	2	no	—
155	C₂²×C₄×C₂₃	—	2	4	no	—
156	C₂²×C₄×C₂₃	—	2	4	no	—
157	C₂²×C₄×C₂₃	—	4	8	no	—
158	C₂²×C₄×C₂₃	—	4	4	no	—
159	C₂²×C₄×C₂₃	—	4	4	no	—
160	C₂²×C₄×C₂₃	C₂×D₄	1	4	yes	—
161	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
162	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
163	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
164	C₂²×C₄×C₂₃	C₂⁴	1	4	yes	—
165	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
166	D₈×C₂₃	—	2	8	no	—
167	M₁₆×C₂₃	—	2	16	no	—
168	C₂×C₈×C₂₃	—	2	4	no	—
169	C₂×C₈×C₂₃	—	2	2	no	—
170	C₂×C₈×C₂₃	#41	1	4	yes	complemented
171	C₂×C₈×C₂₃	C₂⁴	1	4	yes	—
172	C₂×C₈×C₂₃	C₂⁴	1	2	yes	—
173	C₂×C₈×C₂₃	C₂⁴	1	2	yes	—
174	C₄⋊C₄×C₂₃	—	2	1	no	—
175	C₄⋊C₄×C₂₃	—	2	2	no	—
176	C₄⋊C₄×C₂₃	—	2	8	no	—
177	C₄⋊C₄×C₂₃	—	2	4	no	—
178	C₄⋊C₄×C₂₃	—	2	4	no	—
179	C₄⋊C₄×C₂₃	—	2	1	no	—
180	C₄⋊C₄×C₂₃	—	2	2	no	—
181	C₄⋊C₄×C₂₃	C₂×D₄	1	4	yes	—
182	C₄⋊C₄×C₂₃	C₂⁴	1	4	yes	—
183	C₄⋊C₄×C₂₃	C₂⁴	1	2	yes	—
184	C₄⋊C₄×C₂₃	C₂⁴	1	4	yes	—
185	C₄⋊C₄×C₂₃	C₂⁴	1	2	yes	—
186	C₂²⋊C₄×C₂₃	—	2	8	no	—
187	C₂²⋊C₄×C₂₃	—	2	8	no	—
188	C₂²⋊C₄×C₂₃	—	2	4	no	—
189	C₂²⋊C₄×C₂₃	—	2	4	no	—
190	C₂²⋊C₄×C₂₃	—	2	8	no	—
191	C₂²⋊C₄×C₂₃	C₂×D₄	1	8	yes	—
192	C₄²×C₂₃	—	2	8	no	—
193	C₄²×C₂₃	C₂×D₄	1	2	yes	—
194	C₄²×C₂₃	C₂⁴	1	2	yes	—
195	C₂×#3354	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
196	C₂³×C₂₃	—	2	4	no	—
197	C₂³×C₂₃	—	2	4	no	—
198	C₂³×C₂₃	—	2	4	no	—
199	C₂³×C₂₃	—	2	2	no	—
200	C₂³×C₂₃	—	2	2	no	—
201	C₂³×C₂₃	—	4	2	no	—
202	C₂³×C₂₃	—	4	2	no	—
203	C₂³×C₂₃	—	4	2	no	—
204	C₂³×C₂₃	—	4	2	no	—
205	C₂³×C₂₃	—	4	2	no	—
206	C₂³×C₂₃	—	4	2	no	—
207	C₂³×C₂₃	C₂²×D₄	1	4	yes	—
208	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
209	Q₈×C₂₃	—	4	8	no	—
210	Q₈×C₂₃	—	8	4	no	—
211	D₄×C₂₃	—	2	8	no	—
212	D₄×C₂₃	—	2	2	no	—
213	D₄×C₂₃	—	2	2	no	—
214	D₄×C₂₃	—	2	2	no	—
215	D₄×C₂₃	—	2	2	no	—
216	D₄×C₂₃	—	4	16	no	—
217	D₄×C₂₃	—	4	2	no	—
218	D₄×C₂₃	—	4	2	no	—
219	D₄×C₂₃	—	8	8	no	—
220	C₂×C₄×C₂₃	—	2	4	no	—
221	C₂×C₄×C₂₃	—	2	2	no	—
222	C₂×C₄×C₂₃	—	2	8	no	—
223	C₂×C₄×C₂₃	—	2	4	no	—
224	C₂×C₄×C₂₃	—	2	4	no	—
225	C₂×C₄×C₂₃	—	2	2	no	—
226	C₂×C₄×C₂₃	—	2	4	no	—
227	C₂×C₄×C₂₃	—	2	2	no	—
228	C₂×C₄×C₂₃	—	2	4	no	—
229	C₂×C₄×C₂₃	—	4	16	no	—
230	C₂×C₄×C₂₃	—	4	2	no	—
231	C₂×C₄×C₂₃	—	4	2	no	—
232	C₂×C₄×C₂₃	—	4	4	no	—
233	C₂×C₄×C₂₃	—	4	4	no	—
234	C₂×C₄×C₂₃	—	4	8	no	—
235	C₂×C₄×C₂₃	—	4	8	no	—
236	C₂×C₄×C₂₃	AGL(1,ℤ/8)	1	4	yes	—
237	C₂×C₄×C₂₃	C₂²×D₄	1	4	yes	—
238	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
239	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
240	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
241	C₂×C₄×C₂₃	C₂×#41	1	2	yes	—
242	C₂×C₄×C₂₃	2₋¹⁺⁴	1	2	yes	—
243	C₂×C₄×C₂₃	C₂⁵	1	1	yes	characteristic
244	C₈×C₂₃	—	2	4	no	—
245	C₈×C₂₃	—	2	2	no	—
246	C₈×C₂₃	C₂×#41	1	4	yes	complemented
247	C₂×#555	C₂×C₂₃	1	1	yes	characteristic, complemented
248	C₂×#547	C₂×C₂₃	1	1	yes	characteristic, complemented
249	#3354	C₂×C₂₃	1	16	yes	direct product factor
250	C₂×#498	C₂×C₂₃	1	1	yes	characteristic, complemented
251	C₂×#481	C₂×C₂₃	1	2	yes	complemented
252	C₂×#482	C₂×C₂₃	1	2	yes	complemented
253	C₂×#476	C₂×C₂₃	1	1	yes	characteristic, complemented
254	C₂×#474	C₂×C₂₃	1	1	yes	characteristic, complemented
255	C₂×#469	C₂×C₂₃	1	2	yes	complemented
256	C₂×C₄×D₈	C₂×C₂₃	1	1	yes	characteristic, complemented
257	C₂×#436	C₂×C₂₃	1	1	yes	characteristic, complemented
258	C₂×#428	C₂×C₂₃	1	2	yes	complemented
259	C₂²×C₂₃	—	2	4	no	—
260	C₂²×C₂₃	—	2	4	no	—
261	C₂²×C₂₃	—	2	4	no	—
262	C₂²×C₂₃	—	4	8	no	—
263	C₂²×C₂₃	—	4	2	no	—
264	C₂²×C₂₃	—	4	2	no	—
265	C₂²×C₂₃	—	4	2	no	—
266	C₂²×C₂₃	—	4	2	no	—
267	C₂²×C₂₃	—	4	2	no	—
268	C₂²×C₂₃	—	4	2	no	—
269	C₂²×C₂₃	—	8	4	no	—
270	C₂²×C₂₃	—	8	4	no	—
271	C₂²×C₂₃	#547	1	2	yes	—
272	C₂²×C₂₃	C₂×AGL(1,ℤ/8)	1	2	yes	—
273	C₂²×C₂₃	#577	1	2	yes	—
274	C₂²×C₂₃	C₂³×D₄	1	1	yes	characteristic
275	C₄×C₂₃	—	2	4	no	—
276	C₄×C₂₃	—	2	2	no	—
277	C₄×C₂₃	—	4	2	no	—
278	C₄×C₂₃	—	4	2	no	—
279	C₄×C₂₃	—	4	4	no	—
280	C₄×C₂₃	—	4	4	no	—
281	C₄×C₂₃	#547	1	2	yes	—
282	C₄×C₂₃	C₂×AGL(1,ℤ/8)	1	4	yes	—
283	C₄×C₂₃	C₂²×#41	1	1	yes	characteristic
284	C₄×C₂₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
285	C₂²×#41	—	2	1	no	—
286	C₂²×D₈	C₂²×C₂₃	1	1	yes	characteristic
287	C₂²×M₁₆	C₂²×C₂₃	1	2	yes	—
288	#555	—	2	4	no	—
289	#555	C₂²×C₂₃	1	8	yes	complemented
290	#547	—	2	4	no	—
291	#547	C₂²×C₂₃	1	8	yes	complemented
292	C₂×C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
293	C₂×C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
294	C₂×#124	—	2	2	no	—
295	C₂×#123	—	2	2	no	—
296	C₂×C₂²⋊Q₈	—	2	1	no	—
297	C₂×#121	—	2	1	no	—
298	C₂×#121	C₂²×C₂₃	1	2	yes	—
299	C₂×#121	C₂²×C₂₃	1	2	yes	—
300	C₂×C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
301	C₂×C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
302	C₂×C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
303	C₂×#117	—	2	1	no	—
304	C₂×#117	C₂²×C₂₃	1	2	yes	—
305	C₂²×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
306	#498	C₂²×C₂₃	1	8	yes	complemented
307	#482	C₂²×C₂₃	1	16	yes	complemented
308	#481	C₂²×C₂₃	1	16	yes	complemented
309	#476	C₂²×C₂₃	1	8	yes	complemented
310	#474	C₂²×C₂₃	1	8	yes	complemented
311	#469	C₂²×C₂₃	1	16	yes	complemented
312	C₄×D₈	C₂²×C₂₃	1	8	yes	complemented
313	#436	C₂²×C₂₃	1	8	yes	complemented
314	#428	C₂²×C₂₃	1	16	yes	complemented
315	C₂×#107	C₂²×C₂₃	1	2	yes	—
316	C₂×#107	C₂²×C₂₃	1	1	yes	characteristic
317	C₂×#106	C₂²×C₂₃	1	2	yes	—
318	C₂×#106	C₂²×C₂₃	1	2	yes	—
319	C₂×#106	C₂²×C₂₃	1	2	yes	—
320	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
321	C₂×#102	C₂²×C₂₃	1	1	yes	characteristic
322	C₂×#102	C₂²×C₂₃	1	2	yes	—
323	C₂×#102	C₂²×C₂₃	1	1	yes	characteristic
324	C₂×#102	C₂²×C₂₃	1	2	yes	—
325	C₂×C₂²⋊C₈	C₂²×C₂₃	1	2	yes	—
326	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
327	C₂×C₂₃	—	4	4	no	—
328	C₂×C₂₃	—	8	2	no	—
329	C₂×C₂₃	—	8	2	no	—
330	C₂×C₂₃	#3354	1	4	yes	direct product factor
331	C₂×C₂₃	C₂×#547	1	1	yes	characteristic
332	C₂×C₂₃	C₂²×AGL(1,ℤ/8)	1	1	yes	characteristic
333	C₂×C₂₃	C₂×#577	1	1	yes	characteristic
334	C₂×#41	—	2	8	no	—
335	C₂×#41	—	4	8	no	—
336	C₂²×Q₈	—	2	1	no	—
337	C₂²×Q₈	—	2	1	no	—
338	C₂²×D₄	—	2	2	no	—
339	C₂²×D₄	—	2	2	no	—
340	C₂²×D₄	D₄×C₂₃	1	1	yes	characteristic
341	C₂²×D₄	C₂³×C₂₃	1	1	yes	characteristic
342	C₂²×D₄	C₂³×C₂₃	1	1	yes	characteristic
343	C₂³×C₄	—	2	2	no	—
344	C₂³×C₄	—	2	1	no	—
345	C₂³×C₄	—	2	1	no	—
346	C₂³×C₄	C₂³×C₂₃	1	2	yes	—
347	C₂×D₈	—	2	4	no	—
348	C₂×D₈	C₂³×C₂₃	1	4	yes	—
349	C₂×M₁₆	—	2	8	no	—
350	C₂×M₁₆	C₂³×C₂₃	1	8	yes	—
351	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
352	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
353	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
354	C₄⋊Q₈	—	2	2	no	—
355	C₄⋊Q₈	—	2	2	no	—
356	C₄⋊Q₈	C₂³×C₂₃	1	4	yes	—
357	C₄⋊Q₈	C₂³×C₂₃	1	4	yes	—
358	#124	—	2	16	no	—
359	#123	—	2	16	no	—
360	C₂²⋊Q₈	—	2	8	no	—
361	#121	—	2	8	no	—
362	#121	—	2	4	no	—
363	#121	—	2	4	no	—
364	#121	C₂³×C₂₃	1	8	yes	—
365	#121	C₂³×C₂₃	1	8	yes	—
366	C₄×D₄	—	2	2	no	—
367	C₄×D₄	—	2	2	no	—
368	C₄×D₄	D₄×C₂₃	1	4	yes	—
369	C₄×D₄	C₂³×C₂₃	1	4	yes	—
370	C₄×D₄	C₂³×C₂₃	1	4	yes	—
371	C₄×D₄	C₂³×C₂₃	1	4	yes	—
372	#117	—	2	4	no	—
373	#117	—	2	8	no	—
374	#117	C₂³×C₂₃	1	8	yes	—
375	C₂×C₄⋊C₄	—	2	2	no	—
376	C₂×C₄⋊C₄	—	2	4	no	—
377	C₂×C₄⋊C₄	—	2	1	no	—
378	C₂×C₄⋊C₄	—	2	1	no	—
379	C₂×C₄⋊C₄	D₄×C₂₃	1	1	yes	characteristic
380	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
381	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
382	C₂×C₄⋊C₄	C₂³×C₂₃	1	8	yes	—
383	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
384	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
385	C₂×C₂²⋊C₄	—	2	2	no	—
386	C₂×C₂²⋊C₄	—	2	2	no	—
387	C₂×C₂²⋊C₄	—	2	1	no	—
388	C₂×C₂²⋊C₄	—	2	1	no	—
389	C₂×C₂²⋊C₄	—	2	2	no	—
390	C₂×C₂²⋊C₄	D₄×C₂₃	1	2	yes	—
391	C₂×C₄²	—	2	2	no	—
392	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
393	#107	C₂³×C₂₃	1	4	yes	—
394	#107	C₂³×C₂₃	1	8	yes	—
395	#106	C₂³×C₂₃	1	8	yes	—
396	#106	C₂³×C₂₃	1	8	yes	—
397	#106	C₂³×C₂₃	1	8	yes	—
398	C₄⋊C₈	C₂³×C₂₃	1	4	yes	—
399	#102	C₂³×C₂₃	1	4	yes	—
400	#102	C₂³×C₂₃	1	4	yes	—
401	#102	C₂³×C₂₃	1	8	yes	—
402	#102	C₂³×C₂₃	1	8	yes	—
403	C₂²⋊C₈	C₂³×C₂₃	1	8	yes	—
404	C₄×C₈	C₂³×C₂₃	1	4	yes	—
405	C₂₃	C₂×#3354	1	1	yes	characteristic, direct product factor, Sylow
406	C₂⁴	—	2	1	no	—
407	C₂⁴	—	2	1	no	—
408	C₂⁴	C₂×D₄×C₂₃	1	2	yes	—
409	#41	—	4	32	no	—
410	C₂×Q₈	—	2	4	no	—
411	C₂×Q₈	—	2	4	no	—
412	C₂×Q₈	—	4	4	no	—
413	C₂×Q₈	—	8	2	no	—
414	C₂×D₄	—	2	8	no	—
415	C₂×D₄	—	2	8	no	—
416	C₂×D₄	—	2	4	no	—
417	C₂×D₄	—	2	1	no	—
418	C₂×D₄	—	2	1	no	—
419	C₂×D₄	—	2	2	no	—
420	C₂×D₄	—	2	2	no	—
421	C₂×D₄	—	2	2	no	—
422	C₂×D₄	—	2	2	no	—
423	C₂×D₄	—	4	8	no	—
424	C₂×D₄	—	8	4	no	—
425	C₂×D₄	C₂×D₄×C₂₃	1	4	yes	—
426	C₂×D₄	C₂⁴×C₂₃	1	2	yes	—
427	C₂×D₄	C₂⁴×C₂₃	1	2	yes	—
428	C₂²×C₄	—	2	4	no	—
429	C₂²×C₄	—	2	4	no	—
430	C₂²×C₄	—	2	1	no	—
431	C₂²×C₄	—	2	2	no	—
432	C₂²×C₄	—	2	1	no	—
433	C₂²×C₄	—	2	2	no	—
434	C₂²×C₄	—	2	8	no	—
435	C₂²×C₄	—	2	4	no	—
436	C₂²×C₄	—	2	4	no	—
437	C₂²×C₄	—	4	8	no	—
438	C₂²×C₄	—	4	4	no	—
439	C₂²×C₄	—	4	4	no	—
440	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
441	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
442	C₂²×C₄	C₂×D₄×C₂₃	1	4	yes	—
443	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
444	C₂²×C₄	C₂⁴×C₂₃	1	4	yes	—
445	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
446	D₈	—	2	8	no	—
447	M₁₆	—	2	16	no	—
448	C₂×C₈	—	2	4	no	—
449	C₂×C₈	—	2	2	no	—
450	C₂×C₈	#41×C₂₃	1	4	yes	complemented
451	C₂×C₈	C₂⁴×C₂₃	1	2	yes	—
452	C₂×C₈	C₂⁴×C₂₃	1	4	yes	—
453	C₂×C₈	C₂⁴×C₂₃	1	2	yes	—
454	C₄⋊C₄	—	2	1	no	—
455	C₄⋊C₄	—	2	2	no	—
456	C₄⋊C₄	—	2	8	no	—
457	C₄⋊C₄	—	2	4	no	—
458	C₄⋊C₄	—	2	4	no	—
459	C₄⋊C₄	—	2	1	no	—
460	C₄⋊C₄	—	2	2	no	—
461	C₄⋊C₄	C₂×D₄×C₂₃	1	4	yes	—
462	C₄⋊C₄	C₂⁴×C₂₃	1	4	yes	—
463	C₄⋊C₄	C₂⁴×C₂₃	1	2	yes	—
464	C₄⋊C₄	C₂⁴×C₂₃	1	4	yes	—
465	C₄⋊C₄	C₂⁴×C₂₃	1	2	yes	—
466	C₂²⋊C₄	—	2	8	no	—
467	C₂²⋊C₄	—	2	8	no	—
468	C₂²⋊C₄	—	2	4	no	—
469	C₂²⋊C₄	—	2	4	no	—
470	C₂²⋊C₄	—	2	8	no	—
471	C₂²⋊C₄	C₂×D₄×C₂₃	1	8	yes	—
472	C₄²	—	2	8	no	—
473	C₄²	C₂×D₄×C₂₃	1	2	yes	—
474	C₄²	C₂⁴×C₂₃	1	2	yes	—
475	C₂³	—	2	4	no	—
476	C₂³	—	2	4	no	—
477	C₂³	—	2	4	no	—
478	C₂³	—	2	2	no	—
479	C₂³	—	2	2	no	—
480	C₂³	—	4	2	no	—
481	C₂³	—	4	2	no	—
482	C₂³	—	4	2	no	—
483	C₂³	—	4	2	no	—
484	C₂³	—	4	2	no	—
485	C₂³	—	4	2	no	—
486	C₂³	C₂²×D₄×C₂₃	1	4	yes	—
487	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
488	Q₈	—	4	8	no	—
489	Q₈	—	8	4	no	—
490	D₄	—	2	8	no	—
491	D₄	—	2	2	no	—
492	D₄	—	2	2	no	—
493	D₄	—	2	2	no	—
494	D₄	—	2	2	no	—
495	D₄	—	4	16	no	—
496	D₄	—	4	2	no	—
497	D₄	—	4	2	no	—
498	D₄	—	8	8	no	—
499	C₂×C₄	—	2	4	no	—
500	C₂×C₄	—	2	2	no	—
501	C₂×C₄	—	2	8	no	—
502	C₂×C₄	—	2	4	no	—
503	C₂×C₄	—	2	4	no	—
504	C₂×C₄	—	2	2	no	—
505	C₂×C₄	—	2	4	no	—
506	C₂×C₄	—	2	2	no	—
507	C₂×C₄	—	2	4	no	—
508	C₂×C₄	—	4	16	no	—
509	C₂×C₄	—	4	2	no	—
510	C₂×C₄	—	4	4	no	—
511	C₂×C₄	—	4	2	no	—
512	C₂×C₄	—	4	4	no	—
513	C₂×C₄	—	4	8	no	—
514	C₂×C₄	—	4	8	no	—
515	C₂×C₄	C₂₃×AGL(1,ℤ/8)	1	4	yes	—
516	C₂×C₄	C₂²×D₄×C₂₃	1	4	yes	—
517	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
518	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
519	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
520	C₂×C₄	C₂×#41×C₂₃	1	2	yes	—
521	C₂×C₄	C₂₃×2₋¹⁺⁴	1	2	yes	—
522	C₂×C₄	C₂⁵×C₂₃	1	1	yes	characteristic
523	C₈	—	2	4	no	—
524	C₈	—	2	2	no	—
525	C₈	C₂×#41×C₂₃	1	4	yes	complemented
526	C₂²	—	2	4	no	—
527	C₂²	—	2	4	no	—
528	C₂²	—	2	4	no	—
529	C₂²	—	4	8	no	—
530	C₂²	—	4	2	no	—
531	C₂²	—	4	2	no	—
532	C₂²	—	4	2	no	—
533	C₂²	—	4	2	no	—
534	C₂²	—	4	2	no	—
535	C₂²	—	4	2	no	—
536	C₂²	—	8	4	no	—
537	C₂²	—	8	4	no	—
538	C₂²	C₂₃×#547	1	2	yes	—
539	C₂²	C₂×C₂₃×AGL(1,ℤ/8)	1	2	yes	—
540	C₂²	C₂₃×#577	1	2	yes	—
541	C₂²	C₂³×D₄×C₂₃	1	1	yes	characteristic
542	C₄	—	2	4	no	—
543	C₄	—	2	2	no	—
544	C₄	—	4	2	no	—
545	C₄	—	4	4	no	—
546	C₄	—	4	2	no	—
547	C₄	—	4	4	no	—
548	C₄	C₂₃×#547	1	2	yes	—
549	C₄	C₂×C₂₃×AGL(1,ℤ/8)	1	4	yes	—
550	C₄	C₂²×#41×C₂₃	1	1	yes	characteristic
551	C₄	C₂×C₂₃×2₋¹⁺⁴	1	1	yes	characteristic
552	C₂	—	4	4	no	—
553	C₂	—	8	2	no	—
554	C₂	—	8	2	no	—
555	C₂	C₂₃×#3354	1	4	yes	direct product factor
556	C₂	C₂×C₂₃×#547	1	1	yes	characteristic
557	C₂	C₂²×C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
558	C₂	C₂×C₂₃×#577	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₂₃	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₂₃
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₂₃ ◁ C₂×C₄×D₄×C₂₃ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₂×C₂₃×#555	C₂	semidirect
2. C₂×C₂₃×#547	C₂	semidirect
3. C₂₃×#3354	C₂	direct (N × H)
4. C₂×C₂₃×#498	C₂	semidirect
5. C₂×C₂₃×#481	C₂	semidirect
6. C₂×C₂₃×#482	C₂	semidirect
7. C₂×C₂₃×#476	C₂	semidirect
8. C₂×C₂₃×#474	C₂	semidirect
9. C₂×C₂₃×#469	C₂	semidirect
10. C₂×C₄×D₈×C₂₃	C₂	semidirect
11. C₂×C₂₃×#436	C₂	semidirect
12. C₂×C₂₃×#428	C₂	semidirect
17. C₂₃×#555	C₂²	semidirect
19. C₂₃×#547	C₂²	semidirect
34. C₂₃×#498	C₂²	semidirect
35. C₂₃×#482	C₂²	semidirect
36. C₂₃×#481	C₂²	semidirect
37. C₂₃×#476	C₂²	semidirect
38. C₂₃×#474	C₂²	semidirect
39. C₂₃×#469	C₂²	semidirect
40. C₄×D₈×C₂₃	C₂²	semidirect
41. C₂₃×#436	C₂²	semidirect
42. C₂₃×#428	C₂²	semidirect
170. C₂×C₈×C₂₃	#41	semidirect
195. C₂×#3354	C₂₃	direct (N × H)
246. C₈×C₂₃	C₂×#41	semidirect
247. C₂×#555	C₂×C₂₃	semidirect
248. C₂×#547	C₂×C₂₃	semidirect
249. #3354	C₂×C₂₃	direct (N × H)
250. C₂×#498	C₂×C₂₃	semidirect
251. C₂×#481	C₂×C₂₃	semidirect
252. C₂×#482	C₂×C₂₃	semidirect
253. C₂×#476	C₂×C₂₃	semidirect
254. C₂×#474	C₂×C₂₃	semidirect
255. C₂×#469	C₂×C₂₃	semidirect
256. C₂×C₄×D₈	C₂×C₂₃	semidirect
257. C₂×#436	C₂×C₂₃	semidirect
258. C₂×#428	C₂×C₂₃	semidirect
289. #555	C₂²×C₂₃	semidirect
291. #547	C₂²×C₂₃	semidirect
306. #498	C₂²×C₂₃	semidirect
307. #482	C₂²×C₂₃	semidirect
308. #481	C₂²×C₂₃	semidirect
309. #476	C₂²×C₂₃	semidirect
310. #474	C₂²×C₂₃	semidirect
311. #469	C₂²×C₂₃	semidirect
312. C₄×D₈	C₂²×C₂₃	semidirect
313. #436	C₂²×C₂₃	semidirect
314. #428	C₂²×C₂₃	semidirect
330. C₂×C₂₃	#3354	direct (N × H)
405. C₂₃	C₂×#3354	direct (N × H)
450. C₂×C₈	#41×C₂₃	semidirect
525. C₈	C₂×#41×C₂₃	semidirect
555. C₂	C₂₃×#3354	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₂₃×#555	2
C₂×C₂₃×#547	2
C₂₃×#3354	2
C₂×C₂₃×#498	2
C₂×C₂₃×#481	2
C₂×C₂₃×#482	2
C₂×C₂₃×#476	2
C₂×C₂₃×#474	2
C₂×C₂₃×#469	2
C₂×C₄×D₈×C₂₃	2
C₂×C₂₃×#436	2
C₂×C₂₃×#428	2
C₂×#3354	23

Subgroups — C2×C23×#3354 #19208873

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₂₃×#3354 #19208873